Convergence and stabilization of stress-point integration in mesh-free and particle methods

Thomas Peter Fries, Ted Belytschko

Research output: Contribution to journalArticleResearchpeer-review

Abstract

Stress-point integration provides significant reductions in the computational effort of mesh-free Galerkin methods by using fewer integration points, and thus facilitates the use of mesh-free methods in applications where full integration would be prohibitively expensive. The influence of stress-point integration on the convergence and stability properties of mesh-free methods is studied. It is shown by numerical examples that for regular nodal arrangements, good rates of convergence can be achieved. For non-uniform nodal arrangements, stress-point integration is associated with a mild instability which is manifested by small oscillations. Addition of stabilization improves the rates of convergence significantly. The stability properties are investigated by an eigenvalue study of the Laplace operator. It is found that the eigenvalues of the stress-point quadrature models are between those of full integration and nodal integration. Stabilized stress-point integration is proposed in order to improve convergence and stability properties.

Original languageEnglish
Pages (from-to)1067-1087
Number of pages21
JournalInternational journal for numerical methods in engineering
Volume74
Issue number7
DOIs
Publication statusPublished - 14 May 2008

Fingerprint

Meshfree Method
Particle Method
Stabilization
Stability and Convergence
Arrangement
Rate of Convergence
Nodal Integration
Eigenvalue
Galerkin methods
Laplace Operator
Galerkin Method
Quadrature
Oscillation
Numerical Examples

Keywords

  • Convergence
  • Integration
  • Mesh-free method
  • Meshless method
  • Moving least squares
  • Particle method

ASJC Scopus subject areas

  • Engineering (miscellaneous)
  • Applied Mathematics
  • Computational Mechanics

Cite this

Convergence and stabilization of stress-point integration in mesh-free and particle methods. / Fries, Thomas Peter; Belytschko, Ted.

In: International journal for numerical methods in engineering, Vol. 74, No. 7, 14.05.2008, p. 1067-1087.

Research output: Contribution to journalArticleResearchpeer-review

@article{017e3de391b545d6885d93c6fc5379ac,
title = "Convergence and stabilization of stress-point integration in mesh-free and particle methods",
abstract = "Stress-point integration provides significant reductions in the computational effort of mesh-free Galerkin methods by using fewer integration points, and thus facilitates the use of mesh-free methods in applications where full integration would be prohibitively expensive. The influence of stress-point integration on the convergence and stability properties of mesh-free methods is studied. It is shown by numerical examples that for regular nodal arrangements, good rates of convergence can be achieved. For non-uniform nodal arrangements, stress-point integration is associated with a mild instability which is manifested by small oscillations. Addition of stabilization improves the rates of convergence significantly. The stability properties are investigated by an eigenvalue study of the Laplace operator. It is found that the eigenvalues of the stress-point quadrature models are between those of full integration and nodal integration. Stabilized stress-point integration is proposed in order to improve convergence and stability properties.",
keywords = "Convergence, Integration, Mesh-free method, Meshless method, Moving least squares, Particle method",
author = "Fries, {Thomas Peter} and Ted Belytschko",
year = "2008",
month = "5",
day = "14",
doi = "10.1002/nme.2198",
language = "English",
volume = "74",
pages = "1067--1087",
journal = "International journal for numerical methods in engineering",
issn = "0029-5981",
publisher = "John Wiley and Sons Ltd",
number = "7",

}

TY - JOUR

T1 - Convergence and stabilization of stress-point integration in mesh-free and particle methods

AU - Fries, Thomas Peter

AU - Belytschko, Ted

PY - 2008/5/14

Y1 - 2008/5/14

N2 - Stress-point integration provides significant reductions in the computational effort of mesh-free Galerkin methods by using fewer integration points, and thus facilitates the use of mesh-free methods in applications where full integration would be prohibitively expensive. The influence of stress-point integration on the convergence and stability properties of mesh-free methods is studied. It is shown by numerical examples that for regular nodal arrangements, good rates of convergence can be achieved. For non-uniform nodal arrangements, stress-point integration is associated with a mild instability which is manifested by small oscillations. Addition of stabilization improves the rates of convergence significantly. The stability properties are investigated by an eigenvalue study of the Laplace operator. It is found that the eigenvalues of the stress-point quadrature models are between those of full integration and nodal integration. Stabilized stress-point integration is proposed in order to improve convergence and stability properties.

AB - Stress-point integration provides significant reductions in the computational effort of mesh-free Galerkin methods by using fewer integration points, and thus facilitates the use of mesh-free methods in applications where full integration would be prohibitively expensive. The influence of stress-point integration on the convergence and stability properties of mesh-free methods is studied. It is shown by numerical examples that for regular nodal arrangements, good rates of convergence can be achieved. For non-uniform nodal arrangements, stress-point integration is associated with a mild instability which is manifested by small oscillations. Addition of stabilization improves the rates of convergence significantly. The stability properties are investigated by an eigenvalue study of the Laplace operator. It is found that the eigenvalues of the stress-point quadrature models are between those of full integration and nodal integration. Stabilized stress-point integration is proposed in order to improve convergence and stability properties.

KW - Convergence

KW - Integration

KW - Mesh-free method

KW - Meshless method

KW - Moving least squares

KW - Particle method

UR - http://www.scopus.com/inward/record.url?scp=43749088430&partnerID=8YFLogxK

U2 - 10.1002/nme.2198

DO - 10.1002/nme.2198

M3 - Article

VL - 74

SP - 1067

EP - 1087

JO - International journal for numerical methods in engineering

JF - International journal for numerical methods in engineering

SN - 0029-5981

IS - 7

ER -